On times to quasi-stationarity for birth and death processes

The purpose of this paper is to present a probabilistic proof of the well-known result stating that the time needed by a continuous-time finite birth and death process for going from the left end to the right end of its state space is a sum of independent exponential variables whose parameters are the sign reversed eigenvalues of the underlying generator with a Dirichlet condition at the right end. The exponential variables appear as fastest strong quasi-stationary times for successive dual processes associated to the original absorbed process. As an aftermath, we get an interesting probabilistic representation of the time marginal laws of the process in terms of “local equilibria”